Kurt Gödel

Kurt Gödel (1906-1978) was probably the most strikingly original
and important logician of the twentieth century. He proved
the incompleteness of axioms for arithmetic (his most famous result), as
well as the relative consistency of the axiom
of choice and continuum hypothesis with the other axioms of set theory.

Gödel's work was the surprising culmination of a long search for
foundations.
Throughout the nineteenth century, mathematicians had tried to establish
the foundations of calculus. First Cauchy gave the modern definition of
limits; later Weierstrass and Dedekind gave rigorous definitions of the
real numbers. By the end of the century, the foundations of calculus
rested on integers and their arithmetic. This left the problem of putting
the integers themselves on a sound logical basis, which Frege appeared
to solve by defining the positive integers in terms of sets. But it soon
became clear that naive use of sets
could lead to contradictions (such as the set of all sets that aren't
members of themselves). Set theory itself would have to be axiomatized.
In their massive 3-volume Principia Mathematica, Russell and Whitehead
built the foundations of mathematics on a set of axioms for set theory;
they needed hundreds of preliminary results before proving that 1 + 1 = 2.

There remained the problem of analyzing the axioms of set theory.
Mathematicians hoped that their axioms could be proved consistent (free from
contradictions) and complete (strong enough to provide proofs of all true
statements). Gödel showed these hopes were overly naive.
He proved that any consistent formal system strong enough
to axiomatize arithmetic must be incomplete; that is, there are statements
that are true but not provable. Also, one can't hope to prove the
consistency of such a system using the axioms themselves. The basic
idea of Gödel's proof, indirect self-reference, is strikingly simple,
but tricky to grasp. A book-long explanation for the general reader
is offered in Douglas Hofstadter's Gödel, Escher, Bach: An Eternal
Golden Braid.

Gödel was born in Brno, which was then part of the Austria-Hungary.
In 1924 he matriculated at the University of Vienna. He became interested
in logic and was influenced by Hahn, who was to be his thesis advisor.
From 1926-28 he participated in the Vienna Circle that was to become
associated with Rudolf Carnap and logical positivism (though Gödel
disagreed with most of Carnap's views). He completed his dissertation
(on the completeness of first-order logic) in 1929. The next year he
had already proved his incompleteness theorem, and it was published in
1931. (It is ironic that Gödel's first two major results were a
completeness theorem and an incompleteness theorem. The two are not
contradictory, but together they do show that no first-order axiomatization
can capture all the truths of arithmetic). Gödel submitted his
incompleteness paper to the University of Vienna as his
Habilitationsschrift (probationary essay), and in 1933 he was
confirmed as a
Privatdozent: this was not a salaried position, but a certificate
that gave him the right to lecture and collect fees from students.
He taught his first course in the summer of 1933, and that fall he began
a year-long appointment at the newly formed Institute for Advanced Study
(IAS) in Princeton, New Jersey.

Upon his return to Austria the next year, Gödel had the first
of several breakdowns; he spent several months in a sanatorium recovering
from depression. In 1935 he proved the (relative) consistency of the
axiom of choice with the other axioms of set theory. ("Relative" in
this case means that if the axioms other than the axiom of choice
are consistent, then so are these axioms together with the axiom
of choice. As noted above, one can't hope to prove the consistency of
the axioms from themselves.) A second visit to the IAS was cut short
by a relapse of depression, and Gödel remained incapacitated until
spring 1937. Later that year he proved the consistency of the generalized
continuum hypothesis with the axioms of set theory, and he lectured on
his set-theoretic
results at the IAS in 1938-39. By now Austria had been incorporated into
Hitler's Germany, and when he returned home he faced liability for military
service. Though he was not Jewish, Gödel's academic associations
put him in a precarious position. After protracted negotiations he
received a U. S. visa late in 1939: in the early months of the Second
World War he and his wife travelled to the U. S. via the Soviet Union and
Japan. He was given a one-year appointment to the IAS upon
his arrival in Princeton; this was renewed yearly until 1946, when he
was appointed a permanent member.

In 1942 Gödel attempted to prove that the axiom of choice
and continuum hypothesis are independent of (not implied by) the axioms of
set theory. He did not succeed, and the problem remained open until 1963.
(In that year, Paul Cohen proved that the axiom of choice is independent
of the axioms of set theory, and that the continuum hypothesis is
independent of both.) Gödel did little original work
in logic after this, though he did publish a remarkable paper in 1949
on general relativity: he discovered a universe consistent Einstein's
equations in which there were "closed timelike lines"--in such a
universe, one could visit one's own past!

Gödel struck most people as eccentric. His political views
were often surprising: for instance, while he condemned Truman for
fomenting war hysteria and creating the climate for McCarthyism,
he was a great admirer of Eisenhower. While studying for his
U. S. citizenship
examination in 1948, he became convinced he had found an
inconsistency in the Constitution. (Fortunately, this did not
disrupt Gödel's citizenship interview, as the judge brushed aside
the point when Gödel tried to bring it up.)
Gödel became increasingly reclusive in his later years. He was
always somewhat prone to paranoia, was distrustful of doctors, and
tended to feed himself poorly. When his wife was incapacitated with
illness, these factors combined to cause his death from self-starvation.